Acceleration

• Acceleration measures the rate of change of velocity during a given time interval

a = Δv

Δt

• Therefore, the units of m/s or m/s2

s

• Acceleration has a magnitude and a direction, so it is a vector quantity

• Formula for Average Acceleration is:

Use this equation when you are given acceleration,

velocity, and/or time BUT there is NO MENTION of a

displacement!

Ex: As a shuttle bus comes to a stop, it slows from 9.0 m/s to 0.0 m/s in 5 s. Find the acceleration of the bus.

a = Δv/Δt

a = vf – vi/Δt

a = 0.00 – 9.00/5

a = -1.8 m/s2

• You can rearrange the previous equation to solve for an unknown variable; this is how you would rearrange the formula to solve for the final velocity:

Ex: A bus that is traveling at 8.33 m/s speeds up at a constant rate of 3.5 m/s2. What velocity does it reach 6.8 s later?

a = Δv/Δt rearranged to:

a(Δt) + vi = vf

3.5(6.8) + 8.33 = vf

23.8 + 8.33 = vf

32 m/s = vf

Another Acceleration Equation:

xf = xi + vit + ½at2

Use this equation when you are given acceleration, initial

velocity, displacement, and/or time!

Ex: An airplane, initially moving at 3.0 m/s down a runway, begins to accelerate down the runway at 3.6 m/s2. How far down the runway will it be in 20 s?

How to solve…1. List out all known and unknown variables

vi = 3.0 m/s

a = 3.6 m/s2

Δt = 20 s

xf = ?

Hint: unless otherwise stated, we assume that the initial location (xi) is always 0

2. If you do not need to rearrange the equation, then plug the variables into the formula.

xf = xi + vit + ½at2

xf = 0 + 3.0(20) + ½(3.6)(20)2

xf = 0 + 3.0(20) + ½(3.6)(400)

xf = 0 + 60 + 720

xf = 780 m

Last Acceleration Equation:

vf2 = vi

2 + 2aΔx

Use this equation when you are given acceleration, velocity,

and/or displacement BUT there is NO MENTION of time!

Ex: A babysitter pushing a stroller starts from rest and accelerates at a rate of 0.500 m/s2. What is the velocity of the stroller after it has traveled 4.75 m?

How to solve…

1. List out all known and unknown variables

vi = 0 m/s (starts from rest)

a = 0.500 m/s2

vf = ?

Δx = 4.75 m

2. If you do not need to rearrange the equation, then plug the variables into the formula.

vf2 = vi

2 + 2aΔx

vf2 = 02 + 2(0.500)(4.75)

vf2 = 4.75

√vf2 = √4.75

vf = 2.2 m/s

• There may be acceleration problems where we calculate the movement of objects in free fall

• Neglecting air resistance, all freely falling objects (dropped or thrown), fall at the same rate of acceleration i.e., the rate of gravity

• The variable for gravity is g

• g = -9.80 m/s2

*Notice the units of gravity…this should remind you that gravity is a specific acceleration*

Ex: A stone falls freely from rest for 8.0 s. What is the stone’s velocity after 8.0 s?

1. We know:

vi = 0 m/s

vf = ?

t = 8.0 s

a = -9.8 m/s2

2. Given the above info, the formula to use is: a = Δv/Δt

*Note: I chose this formula b/c it was the only one without a displacement*

3. I rearranged the formula to solve for the final velocity: vf = a(Δt) + vi

vf = a(Δt) + vi

vf = -9.8(8.0) + 0

vf = -78.4 m/s

*Note: negative velocity accounts for direction of motion (downwards)

Acceleration Graphs:Acceleration Graphs:Velocity vs. TimeVelocity vs. Time

Acceleration Graphs:Acceleration Graphs:Velocity vs. TimeVelocity vs. Time

What’s going on during each segment?